Stochastic Processes and Their Applications

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1) Herein, we postulate that the glacial cycles can be completely explained by carbon transfer between atmospheric, stored and buried states. This theory builds upon the glacial burial hypothesis as well as the greenhouse gas effect. We support our hypothesis by developing a closed stochastic model that can statistically reproduce the observed ice-core and ocean-sediment samples over the past two million years.We then use this model to explain: the relatively steeper rise out of glacial states as compared to recession into, the current extended Holocene interglacial epoch, and the somewhat mysterious change from the ancient 41ka world of shallower, higher frequency glacial cycles to the current 100ka world of deep cycles. 2) We will obtain representations for superprocesses (Measure-valued Markov processes here) beneficial to analysis and simulation alike. Superprocesses, in filtering theory, branching processes and population genetic, are a high-density, high-activity limit of particle processes. However, the limit looses: a) the particle representation and b) the joint distribution information (over different times). Kurtz and collaborators (Ocone, Donnelly, Xiong, Rodrigues) introduced levels and Markov mappings that allow an infinite collection of particles and their genealogy in the limit. However, this method is still more clever ideas than a unified theory. Separately, Billingsley studies the existence of probability measures corresponding to joint distributions but nobody has addressed the question: When is a superprocess the projection of a random measure on pathspace? We will investigate both representations and determine when there is no single pathspace random measure. 3) The spine decomposition and the representation approach are not panacea for strong laws of large numbers (SLLN) and large deviation principles (LDP) as many superprocesses lack the compact support property and stochastic equation representation respectively. In the 45 years since Watanabe's classical SLLN for branching Markov processes only isolated SLLN for superprocesses over Euclidean space been discovered. We will investigate SLLNs, CLTs, LDPs and LILs for superprocesses in terms of time and particle approximation.